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\begin{center}
{\Large ASTR450 Homework \# 9 -- Perturbations\\
Due Tuesday, April 22}
\end{center}


1. {\bf Impulse Approximation to a Drag Force.} Consider a satellite
on an inclined elliptical orbit acted on by a drag force of the form
${\bf F} = -k{\bf v}_{rel}$, where ${\bf v}_{rel}$ is the velocity of
the satellite relative to the atmosphere, and $k$ is a positive
constant. Recall that Earth's atmosphere decays exponentially with
height (scale height $\approx 10$km).\\ a) Approximate the range of
eccentricities for which the impulse approximation will be
reasonable.\\ b) Consider a rotating Earth.  Start by making a
qualitative estimate of the error made in neglecting rotation. How
does the Earth's rotation affect an equatorial orbit ($i=\Omega=0$)?
Describe how the orbital elements $a, e, i, \Omega, \comega$ vary in
time.\\ c) Now imagine orbits with $i\ne 0, \Omega \ne 0$. Using the
perturbation equations and other physical arguments, describe
qualitatively how these orbits will evolve (i.e.  how $a, e, i,
\Omega, \comega$ vary in time).

2. {\bf Radial Perturbation Forces.} Consider a radial perturbation
force of the form ${\bf F} = R \hat r$.\\ a) Apply the perturbation
equations to this force and obtain simplified expressions for $da/dt$,
$de/dt$, $di/dt$, $d\Omega/dt$, and $\comega/dt$.\\ b) A radial
perturbation to gravity, which is itself a radial force, is an example
of a central force. So angular momentum must be conserved. Show that
your equations conserve the angular momentum vector and describe the
constraints that this imposes on these orbits. \\ c) Now let $R=Ar^n$
where $A$ is a constant. Take the time average of your expressions
over a single unperturbed Keplerian orbit (this step assumes that the
perturbation is small). Show that $<r^n\sin\nu> = 0$ and argue, on
physical ground, that $<\cos\nu>$ is negative (or zero) and that
$<r^{-2}\cos\nu> = 0$. It can be shown that $<r^n\cos\nu>$ is negative
for $n>-2$ and positive for $n<-2$. Use this fact to determine how the
sign of your time-averaged $d\comega/dt$ depends on $A$ and $n$. Use
the Central Force Integrator to check your results numerically.\\ d)
Finally, consider the General Relativistic (GR) Perturbation $R = A
r^{-4}$ where $A$ is a small negative constant.  The integral
$<r^{-4}\cos\nu> = a^{-4}e(1-e^2)^{-5/2}$.  Solve, analytically, for
the value of $A$ that will give 30 degrees of precession per orbit for
$e=0.5$ (The true effects of GR on Mercury's orbit are almost exactly
a million times weaker). Convert your prediction into the proper
initial conditions for the Central Force Integrator, and test it!
Start your orbit at pericenter.


3. {\bf The Three Body Problem.} Use the Three-Body Integrator to
investigate tadpole (paths which surround one of $L_4$ and $L_5$),
horseshoe (paths which orbit both $L_4$ and $L_5$), and more
complicated orbits! First, run through the Three-Body form and the
graphics driver using the default values to get oriented. What do you
think this tadpole orbit will look like in inertial coordinates? Find
out by toggling to inertial coordinates on the graphics driver! Now
you are ready to start. Take the default settings on the Three-Body
form and set the velocities to zero. Your missions, should you choose
to accept them, are below; each of you have a different set of initial
conditions to investigate.  Classify your orbits as either i) tadpole,
ii) horseshoe, iii) entirely inside Jupiter's orbit, iv) entirely
outside Jupiter's orbit, or v) a path which switches between several
of these.  Explore! \hspace{10cm} (over)

\begin{center}
Chris: set y=0, and vary x from 0 to 3 relative to $L_4$\\
Ray: set y=0, and vary x from 0 to -3 relative to $L_4$\\
Elbert: set x=0, and vary y from 0 to  3 relative to $L_4$\\
Alaa: set x=0, and vary y from 0 to -3 relative to $L_4$\\[5pt]
Mike: set y=0, and vary x from 0 to 3 relative to $L_5$\\
Jason: set y=0, and vary x from 0 to -3 relative to $L_5$\\
Linda: set x=0, and vary y from 0 to  3 relative to $L_5$\\
Lori: set x=0, and vary y from 0 to -3 relative to $L_5$\\
\end{center}

Plot -- and think about -- your orbits in both the rotating and
inertial frames. Try to identify and understand the correspondence
between structures seen in one frame and their representations in the
other frame. Using color to represent time evolution can help you
figure out what is going on! Here are some questions to think about
which we will discuss in class. As you increase the distance from the
Lagrange Point, does the region of horseshoe orbits change to a region
of more complicated orbits in a smooth or chaotic way? Can you explain
why? How about transitions between tadpole and horseshoe orbits, and
between other types of orbits? What is the relationship between your
orbits and the Zero Velocity Curves?

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